theorem Th23:
  for G being addGroup, A being Subset of G holds (for g being
  Element of G st g in A holds -g in A) implies -A = A
proof
  let G be addGroup, A be Subset of G;
  assume
A1: for g being Element of G st g in A holds -g in A;
  thus -A c= A
  proof
    let x be object;
    assume x in -A;
    then ex g being Element of G st x = -g & g in A;
    hence thesis by A1;
  end;
  let x be object;
  assume
A2: x in A;
  then reconsider a = x as Element of G;
A3: x = --a;
  -a in A by A1,A2;
  hence thesis by A3;
end;
